What Is Pascal's Triangle? (Part 1)

At the end of the last episode on how to calculate probabilities, I assigned you a little project about flipping 1, 2, 3, and finally 4 coins at once. Your first goal was to use a probability tree—or any other method you like—to figure out how many possible outcomes there are in each case. And your second goal was to figure out how many of these outcomes will give you 0, 1, 2, 3, or 4 heads.

Why would you want to do that? I know this might sound kind of strange, but it’s because the answer you get is sort of surprising…and it’s very cool. Plus, it’s related to a famous and fascinating pattern you may have heard of called Pascal’s triangle. What exactly is that? And where does it come from? Stay tuned, because that’s precisely what we’re talking about today..

Tossing 1 and 2 Coins

The first thing we need to do on our quest to discover Pascal’s triangle is figure out how many possible outcomes there are when tossing 1 and 2 coins at the same time. Of course, when we toss a single coin there are exactly 2 possible outcomes—heads or tails—which we’ll abbreviate as “H” or “T.” How many of these outcomes give 0 heads? Well, 1 of them. And there’s also 1 outcome that gives 1 head.

Thinking about 1 coin is almost too easy, so let’s move on to 2 coins. As we saw when we first learned about probability and probability trees, there are 4 possible outcomes when tossing 2 coins. That makes perfect sense since each of the 2 possible outcomes for the first coin—H or T—has 2 possible outcomes for the second coin—again, H or T. So there must be twice as many outcomes for 2 coins as there are for 1 coin. And there are! HH, HT, TH, or TT.

How many of these outcomes have 0 heads? Only 1: the outcome TT. How about 1 head? There are 2 of those: HT or TH. And 2 heads? Again, we’re back to 1 possible outcome: HH.

Tossing 3 and 4 Coins

Let’s try tossing 3 coins at once. There’s a bit more to keep track of here, so it’s helpful to draw a probability tree. The tree for 3 coins looks exactly like the tree for 2 coins, with the addition of another set of branches on the right side representing the possible outcomes for the third coin. If you draw this probability tree, you’ll find that there are a total of 8 possible outcomes.

That number of outcomes makes sense since, again, there must be twice as many possible outcomes for 3 coins as there are for 2 coins. How many of these outcomes have 0, 1, or 2 heads? If you add them all up, you’ll find 1 outcome with 0 heads (TTT), 3 outcomes with 1 head (HTT, THT, or TTH), 3 outcomes with 2 heads (HHT, HTH, and THH), and finally 1 outcome with 3 heads (HHH).

Four coins is even more complicated since there are now 16 possible outcomes. I’ll spare you the pleasure of having me describe them all right now, but if you draw up the tree diagram and add up the possibilities, you’ll find 1 outcome that gives 0 heads, 4 outcomes that give 1 head, 6 that give 2 heads, back to 4 that give 3 heads, and again 1 outcome that gives 4 heads.